Modern operating systems use cubic Bézier curves extensively to represent many user interface components. For example, the individual glyphs of a font are typically represented as a sequence of cubic Bézier curves. Operating system must therefore contain functionality for generating cubic Bézier curves in order to display textual interfaces and graphical interfaces.
Generally, a cubic Bézier curve is defined by a start point (x0, y0), an end point (x3, y3), and two control points (x1, y1) and (x2, y2). The two control points determine the shape of the curve. Two cubic equations define the points on the curve. Both equations are evaluated for an arbitrary number of values of t between 0 and 1. One equation provides values for x, and the other provides values for y. As values for t are supplied to the equations, the points (x(t), y(t)) define the line between the start point and the end point. Exemplary cubic Bézier curve equations are illustrated below.x(t)=axt3+bxt2+cxt+x0y(t)=ayt3+byt2+cyt+y0The values of the coefficients ax, bx, cx, ay, by, and cy can be determined using the four known points.